Let us first introduce some notation. Let \({P}_{n}\)  (resp. \({P}_{\infty }\) ) be the set of partitions of \([n]\)  (resp. \(\mathbb {N}\) )
endowed with their usual topology
(Lemma 2.6 in [1]}),
and \(\mathbf {0}_n\in {P}_{n}\)  (resp. \(\mathbf {0}_\infty \) ) be the partition of singletons. For a partition \(\pi \) 
let \((\pi _1,\pi _2,\cdots )\)  be the blocks of \(\pi \)  ordered by their least elements, i.e.
